1. Field of the Invention
The present invention is related to the field of digital signal processing. More specifically, the present invention is related to a method of digitally sampling a signal so that frequency components are preserved in the sampled signal at a frequency above the fold-over (Nyquist) frequency determined by the digital sampling interval.
2. Discussion of the Related Art
Digital signal processing, particularly of electrical signals corresponding to physical phenomena such as acoustic amplitudes, typically includes the step of digitizing the electrical signals. Digitizing is to convert a signal into a series of numbers representing instantaneous amplitudes of the signals sampled at spaced apart time intervals. Preserving substantially all the information contained in the signals requires that the spaced apart time intervals be small enough to adequately sample substantially all the frequencies contained in the signals.
A method of determining the spaced apart interval which enables sampling of substantially all the frequencies contained in the signals is known in the art. The method is typically defined by a relationship referred to in the Shannon Sampling Theorem. The Shannon Theorem states, among other things, that the maximum frequency contained in the signals which can be preserved during digital sampling is equal to half the sampling frequency. Expressed in terms of the spaced apart time interval, or sample interval, the theorem states the relationship: ##EQU1##
.DELTA.t in this relationship represents the sample interval, and .function..sub.max is the maximum frequency which can be sampled.
If a particular signal is sampled at too long a sample interval (too low a frequency) for the frequencies which are contained in the particular signal, the resulting series of numbers may not faithfully represent the particular signal because of the presence of inadequately sampled high-frequency content. The presence of inadequately sampled high frequency content typically manifests itself as improperly large amplitudes of some lower frequency components in the digitized signal. This effect is called aliasing.
One method known in the art tier avoiding aliasing is to limit the upper frequency content of the signal which is actually digitized. Analog low pass (high cut) filters are typically interposed between the signal source and an analog to digital converter to limit the uppermost frequency content of the signal being digitized. One of the problems with analog low pass filters is that the filter response may be such that the input signal is only gradually attenuated by the filter as the frequency increases. In order to have adequate attenuation of signal components at or above the fold-over frequency, the low pass filter typically begins attenuating the signal at about 75 percent of the fold-over frequency. Adequate attenuation typically is defined as about 40-60 dB reduction in amplitude. Because of this characteristic of the analog low pass filter, significant signal information can be lost in the frequency range of 75 to 100 percent of the fold-over frequency.
Another method known in the art for avoiding aliasing is to reduce the sample interval so that higher frequencies can be adequately sampled from the particular signal. As the sample interval is reduced and the sampling frequency thereby increased, the corresponding cut-off frequency of the analog low pass filter can be proportionately increased.
A drawback of reducing the sample interval is that the number of samples in the series of numbers, and therefore the volume of digital data, is directly proportional to the sampling frequency. In some applications, such as geophysical exploration, increasing the sampling frequency can be difficult because of the huge volume of digitized signal data which is generated.
It is an object of the present invention to provide a method of digitally sampling signals which adequately samples frequencies in the signal which are above the fold-over frequency.